To sum up, a classic piece? (5)
I believe the answer is:
music
'piece?' is the definition.
(I know that composition is a type of music)
'to sum up a classic' is the wordplay.
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'a classic' becomes 'ic' (I am not sure about this - if you are sure you should believe this answer much more).
'sum' backwards is 'mus'.
'mus'+'ic'='MUSIC'
Can you help me to learn more?
(Other definitions for music that I've seen before include "Strains" , "Organised notes (that must sometimes be faced)" , "Melodious sounds" , "Composed sound" , "Art of instrumental performance" .)